Influences of Non-Locality on the Elastic Wave Surfaces in Elastic Media

Abstract

Classical continuum theories restrict the response of the continuum stringently to local actions, thus these theories are not capable to explain some phenomena precisely where the length scales are often sufficiently short as in nanostructures where it is required to consider the small length scales. This paper within the framework of nonlocal elasticity is concerned with the study of wave-surface features in nonlocal elasticity for cubic crystals. The nonlocal Christoffel equation of wave motion is derived and dispersion relations are obtained. The present model predicts some notable features of the dispersion relations in cubic crystals in comparison with classical local model. By considering the wave and slowness surfaces in [100], [110], and [111] planes of cubic crystals a perceptible change is observed with nonlocality parameter. In nonlocal theory longitudinal and transverse waves, become dispersive and influenced by non-locality parameter, whereas theses waves are non-dispersive in its counterpart classical theory (local theory). It is found that phase and group wave velocities for longitudinal and transverse modes are influenced by the nonlocality parameter only when its value is greater than 0.001. Numerical calculation for crystals Silicon (Si), Aluminum (Al), Copper (Cu), Nickel (Ni), Gold (Au) are carried and found that velocities of longitudinal and transverse waves continuously decreases with increases of non-locality parameter. Polar diagram and wave’s surfaces for phase and group velocities (m/s) of longitudinal and transverse and slowness surfaces are represented graphically in nonlocal elasticity.